Speed work without the work

I have been a runner for years now. Almost 10 to be exact. Photo evidence of my first race and personal running mantra:

Also, Tim looks 12, but, moving on...

I have never really been fast, at the height of my best training year a few seasons ago (before kid #2), I ran a consistent 10 minute mile. My 5K PR is 29 and some change. Now that I am not running a full, I've decided to get faster!

I am currently run/walking, so my pace is particularly slow. This is an average run/walk for me:

This, by contrast, is a recent "all running" 2 miler.

I've never been a huge fan of run/walk for shorter distances because, let's face it, I CAN run that, I'm just choosing not to. 

So my first step in non-work Speed Work is to just run. Anything 4 miles and under is all running. From there, I think quantity of runs could do wonders for my speed! 

Thoughts?